I want to discuss some character formulas that arise in Lie theory and symmetric functions, and how they are related to certain equivariant resolutions. A weird application of these resolutions is that certain partial alternating sums of symmetric functions are Schur positive. It’s unclear to me what that might imply. At first we’ll work in the general Kac–Moody setting, but one can imagine the example if that helps.
Weyl–Kac character formula.
Let be a complex Kac–Moody algebra associated to a Cartan matrix , together with a Cartan subalgebra and Borel subalgebra , and Weyl group . Let be the opposite Borel and set and . The simple coroots are and we define by the condition . We’ll denote the set of dominant weights by and the set of positive roots by . Given a positive root , let be the dimension of the -root space of . Also define the dotted action of on the set of weights via .
Given a dominant weight , we have a unique irreducible representation of with highest weight . Its character is the formal linear combination where is the -weight space. More compactly, this is given by the Weyl–Kac formula:
I’ve written it in this way to suggest that the sign might be coming from the Euler characteristic of some complex. In fact, it does, which is what we will discuss next.
Given any integral weight , we can form a highest weight module as , where is the universal enveloping algebra of , and is the left ideal generated by and for all . This is known as the Verma module, and is a free module over , so that its character is
The Bernstein–Gelfand–Gelfand (BGG) resolution for is the -equivariant resolution
where . Note also that by looking at the -weight space of this complex, we also get Kostant’s multiplicity formula (which makes sense when )
where is the number of ways to write as a sum of positive roots.
The construction for this resolution requires a few facts. First, given a dominant weight , we have that . Furthermore, the dimension is 1 if and only if in the Bruhat order, and all nonzero maps are injective. Hence we may fix an inclusion for all .
So for the map , we only need to pick a constant for the maps when . There is no unique way to do this, but one such way comes from the following two facts. First, given such that , either , or the interval consists of 4 elements. Second, there exists a way to assign the numbers to the covering relations in the Bruhat order of such that in the interval above, the product over the two maximal chains have opposite signs. Then we set according to how these signs are prescribed.
From these two facts, it is then clear that the maps just defined give us a complex. The exactness is more delicate, and I’ll refer the reader to Chapter IX of Kumar’s book Kac–Moody Groups, their Flag Varieties and Representation Theory for more details.
For the second part of this post, I want to discuss a functor that Zelevinsky introduced in “Resolvents, dual pairs, and character formulas” that transforms the BGG resolution into other resolutions that realize the Jacobi–Trudi identity and its variants. A special case of this functor was considered by Akin in “On complexes relating the Jacobi-Trudi identity with the Bernstein-Gel’fand-Gel’fand resolution” for the Jacobi–Trudi identity. In fact, his construction for this case gives something more general, but I won’t discuss it. I won’t be needing to denote the Weyl group anymore, so I’ll use it to name vector spaces (sorry!)
The functor goes from representations of to vector spaces, and has two parameters. The first is a representation of , and the second is a weight . It is defined as
The main properties are that the resulting complex obtained from applying to a BGG resolution is exact, and that
and denotes a nonzero vector in the -root space of and is the Killing form. An important property of this subspace is that its dimension is the multiplicity of in the tensor product .
Note that given the nature of the function , if carries commuting actions of and some other algebra , then the functor goes to representations of . Let’s give some examples.
Let and be two vector spaces. Let . The Cauchy identity gives the -equivariant decomposition
where is the irreducible representation of of highest weight and similarly for . We take and apply the functor to the BGG resolution of . We need to calculate and to figure out what the resulting resolution is.
For the first one, let’s calculate more generally. Since we’re only interested in the action of , we can write where and is the basis that diagonalizes . For an -tuple of nonnegative integers , is the 1-dimensional representation of with character , and . Then we have
so . Finally, what is ? Well, from what we mentioned above, is the multiplicity of in , which is the Littlewood–Richardson coefficient . Hence
where is a skew Schur functor, and the last equality follows from standard properties of skew Schur polynomials.
So the resulting resolution we get from applying to the BGG resolution of looks like where
(). In this case, . Taking the -equivariant Euler characteristic, we get
which is the Jacobi–Trudi identity. If we replace the symmetric powers above with exterior powers (and use the dual Cauchy formula), we’ll get the dual Jacobi–Trudi identity.
Determinantal expression for products of Schur functions.
A slight variation gives a not as well known determinantal formula: we take as above, but apply to the BGG resolution of where . Then is the multiplicity of in , which is equal to
The space of invariants is unchanged by taking a dual, so we see that this dimension is the multiplicity of in , or the Littlewood–Richardson coefficient , so
and hence we’re now resolving a tensor product. This time the -equivariant Euler characteristic is
Note that the product is equal to a skew Schur function by making a shape where the diagrams of and don’t have any boxes in the same row or column, but this will in general have rows, so this determinantal expression is more efficient.
For an extra formula, take with the action of and the action of on the tensor factors. Then Schur–Weyl duality gives
where index irreducible representations of . Apply to the BGG resolution of over and see what you get.